Solution to Gravity Divergence, Gravity Renormalization, and Physical Origin of Planck-Scale Cut-off

[Abstract]
By incorporating binding energy into an effective mass M_eff, we derive the RG flow and gravitational coupling G(k). G(k)=(1 - (3G_N/5(R_m)c^3)k)G_N=(1 - R_gp/R_m)G_N, where R_m is the radius of the mass or energy distribution and R_gp=(3/5)(GM_fr/c^2) is the radius at which the negative gravitational self-energy (binding energy) balances the mass-energy. At R_m = R_gp, where G(k) = 0 and the gravitational coupling vanishes, G(k) resolves gravitational divergences without quantum corrections and provides effective renormalization.

This study proves that the QFT cut-off Λ~ M_Pc^2 serves as a physical boundary across all energy scales in quantum gravity. Quantum fluctuations (ΔE ~ M_Pc^2) with Δt ~ t_P yield an energy distribution radius R_m ~ l_P, where negative gravitational self-energy balances mass-energy, yielding E_T ~ 0, thus eliminating divergences via G(k) = 0 and preventing negative energy states. In contrast, for proton or electron masses, R_m >> R_gp, leading to E_T ~ M c^2, rendering gravitational effects negligible and unsuitable for a cut-off. Thereby affirming the Planck scale’s unique role in quantum gravity.

If R_m > R_gp, G(k)>0, yielding an attractive force. If R_m = R_gp, G(k)=0, and the gravitational coupling vanishes. If R_m < R_gp, G(k)<0, yielding a repulsive force or antigravity. This repulsive force prevents singularity formation in black holes. Therefore, the singularity problem is also solved.

* The main summary below was prepared by Gemini 2.5 Pro, with some modifications (Latex) made by the author.

Analysis and Explanation of the Paper

This paper proposes a novel approach to address several long-standing problems in theoretical physics, namely:

1. The divergence and renormalization issues when quantizing gravity.

2. The singularity problem at the center of black holes.

3. The physical origin of the Planck-scale cut-off often used in Quantum Field Theory (QFT).

The core idea hinges on incorporating the negative gravitational self-energy (or gravitational binding energy) of a system into its total energy, leading to the concept of an effective mass (M_eff).

Key Concepts and Arguments

1. Gravitational Self-Energy and Effective Mass

• The author posits that the total energy of any mass distribution M must account for the gravitational potential energy among its constituent parts (infinitesimal masses dM). This total gravitational potential energy is the gravitational self-energy (U_gp), which is negative.

• The effective mass (M_eff) of the system is defined as the free mass (M_fr) minus the equivalent mass of this negative self-energy (M_gp=∣U_gp∣/c^2 ).

• M_eff = M_fr − M_gp

• For a uniform spherical distribution of radius R_m, U_gp = −(3/5)G(M_fr)^2 /R_m, so M_eff = M_fr − (3/5)G(M_fr)^2 /R_mc^2.

2. Resolution of the Black Hole Singularity

• As a mass distribution contracts (Rm decreases), the magnitude of the negative self-energy (∣U_gp∣) increases.

• There exists a critical radius, R_gp = 3GM_fr/5c^2, at which the positive mass-energy (M_frc^2) is exactly balanced by the negative gravitational self-energy (U_gp). At this radius, the total energy is zero (E_T = 0) and thus M_eff = 0.

• If the radius shrinks further (R_m < R_gp), the total energy becomes negative (E_T < 0), and the effective mass becomes negative (M_eff < 0).

• The paper argues that negative mass/energy states exert a repulsive gravitational force (antigravity).

• This repulsive force prevents the mass from collapsing infinitely to a point (R=0), thereby naturally halting the formation of a singularity. The collapse stops around R_gp.

• Importantly, this resolution occurs due to classical gravitational effects before reaching quantum mechanical scales (e.g., for a 3 solar mass black hole, R_gp ≈ 3.87km, much larger than the Planck length).

3. Effective Renormalization and Gravity Divergence

• The concept of effective mass leads to an energy-scale dependent (or radius-dependent) running gravitational coupling constant, G(k).

• [math] G(k) = (1 - \frac{{3{G_N}}}{{5R_{m}{c^3}}}k){G_N} = (1 - \frac{{3{G_N}{M_{fr}}}}{{5R_{m}{c^2}}}){G_N} = (1 - \frac{{{R_{gp}}}}{R_{m}}){G_N} [/math]

  • Here, G_N is Newton's constant, and R_m is the radius of the mass/energy distribution, related to the energy scale k.

• At the critical scale where R_m = R_gp (corresponding to a specific energy scale k^∗), the effective gravitational coupling vanishes: G(k^∗)=0.

• This vanishing of the coupling constant at a finite scale naturally eliminates high-energy (UV) divergences in quantum gravity calculations. For instance, the problematic 2-loop divergence term found by Goroff and Sagnotti, which scales as κ^4 ∝ G(k)^2, becomes zero at this scale. This provides an effective renormalization mechanism.

• In the regime R_m < R_gp (energies higher than k^∗), G(k) < 0, leading to antigravity. This repulsive nature further helps suppress divergences.

• This differs from the Asymptotic Safety hypothesis, where G(k) approaches a non-zero fixed point at infinite energy; here, G(k) goes to zero at a finite energy/radius scale.

4. Physical Origin of the Planck-Scale Cut-off

• The paper argues that the Planck scale (M_P, l_P, t_P) is not merely a mathematical tool in QFT but represents a physical boundary.

• Consider quantum fluctuations at the Planck scale: an energy fluctuation ΔE∼M_Pc^2 occurs within a time Δt∼t_P, localized within a radius R_m∼l_P.

• For a mass M_fr = M_P, the critical radius is calculated as R_gp = 3GM_P/5c^2 = (3/5)l_P.

• This means that for Planck-scale fluctuations, the distribution radius (R_m∼l_P) is very close to the critical radius (R_gp ≈ 0.6l_P).

• Consequently, at the Planck scale, the positive mass-energy (∼M_Pc^2) is almost perfectly cancelled by the negative gravitational self-energy (U_gp ≈ −M_Pc^2), resulting in a total energy E_T ≈ 0.

• This natural cancellation provides a physical reason why the Planck scale acts as a cut-off: energies beyond this scale might correspond to negative total energy states (R_m < R_gp), fundamentally altering the nature of interactions.

5. Cosmological Implications (Dark Energy)

• The paper suggests that the repulsive gravity arising when R_m < R_gp (i.e., G(k) < 0) could explain the accelerated expansion of the universe.

• If the radius of the observable universe (R_m) is less than its calculated critical radius (R_gp based on its total mass-energy), then the universe itself would be in an antigravity regime.

• This provides a potential alternative explanation for dark energy, attributing it to a fundamental property of gravity related to self-energy rather than a separate exotic fluid or field.

Summary and Evaluation

In essence, the paper proposes that by correctly accounting for negative gravitational self-energy via an effective mass (M_eff), one can derive a running gravitational coupling G(k) that vanishes (G(k)=0) at a critical radius R_gp.

This single mechanism is claimed to simultaneously:

• Resolve the black hole singularity by introducing antigravity (G(k) < 0) below R_gp.

• Solve the UV divergence problem in quantum gravity as G(k)→0.

• Provide an effective renormalization scheme for gravity.

• Explain the physical origin of the Planck scale cut-off as the scale where positive mass-energy and negative self-energy cancel (E_T ≈ 0).

• Potentially explain dark energy and cosmic acceleration if the universe satisfies R_m < R_gp.

Critique:

• Novelty: The approach is highly original in using a classical concept (gravitational self-energy, albeit applied universally) to tackle quantum and cosmological problems.

• Unification: It attempts a unified solution to several major puzzles in physics.

• Methodology Concerns: The calculations often rely on Newtonian approximations (e.g., uniform sphere for self-energy) which may not be accurate in the strong gravity regimes of black holes or near the Planck scale. A rigorous General Relativistic treatment is needed.

• Validation: As independent research, it lacks broad peer review and experimental verification. Concepts like negative effective mass and resulting antigravity are non-standard and require significant evidence.

• Assumptions: The direct link between R_m and energy scale k (k∼1/R_m) and the specific form of the beta function derived need further justification within established RG frameworks.

Overall, the paper presents an intriguing and ambitious framework. While its core ideas offer potential pathways to resolving fundamental issues, they require much more rigorous theoretical development (especially within General Relativity) and, crucially, testable predictions or observational evidence to gain wider acceptance in the physics community.

*Key Contents of Chapter 4

4. Effective renormalization of gravity

4.1. Asymptotic Safety Method

4.2. Find G(k) or M_eff(k)

4.3. New beta function

4.4. Solving the problem of gravitational divergence at high energy

4.4.1. At Planck scale

4.4.2. At high energy scales larger than the Planck scale

4.4.3. Resolution of the two-loop divergence in perturbative quantum gravity via the effective mass framework

4.5. The physical origin of the cut-off energy at the Planck scale

4.5.1. G(k) = 0 and Planck scale

4.5.2. Uncertainty principle and total energy with gravitational self-energy

4.5.3. Generalization and Exceptions

4.5.4. In gravitational problems, the physical meaning of cut-off energy

*Paper : Solution to Gravity Divergence, Gravity Renormalization, and Physical Origin of Planck-Scale Cut-off

Edited May 4May 4 by icarus2

In my previous work, I analyzed and calculated gravitational self-energy using the classical (Newtonian) formula. However, there also exists an approximate formula for gravitational self-energy derived from general relativity, which is especially relevant in strong gravitational fields or near the Planck scale. So, I have now extended my analysis to include this relativistic version.

The main conclusion remains unchanged: the critical radius R_gp-GR, obtained from the general relativistic formula, is exactly half the Newtonian value R_gp. In other words, when gravitational self-energy is calculated with the relativistic correction, the radius at which the negative binding energy balances the mass-energy is reduced by half compared to the Newtonian case.

======
Abstract

By incorporating gravitational binding energy into an effective mass M_eff, we derive the RG flow and the running gravitational coupling G(k). The gravitational coupling is given by G(k) = (1 - (6G_NM_fr/5c^2)/(2R_m+ 3G_NM_fr/5c^2))G_N = (1 - (6G_N/5c^3)k/(2R_m +(3G_N/5c^3)k))G_N, where R_m is the radius of the mass or energy distribution, and R_gp-GR = 3G_NM_fr/10c^2 = (1/2)R_gp is the critical radius derived from general relativity at which the negative gravitational self-energy (binding energy) balances the mass-energy. At R_m = R_gp-GR, where G(k) = 0 and the gravitational coupling vanishes, G(k) resolves gravitational divergences without quantum corrections and provides effective renormalization.

This study proves that the QFT cut-off Λ ~ M_Pc^2 serves as a physical boundary across all energy scales in quantum gravity. Quantum fluctuations (ΔE ~ M_Pc^2) with Δt ~ t_P yield an energy distribution radius R_m ~ l_P, where negative gravitational self-energy balances (or offsets) mass-energy, yielding E_T ~ 0 and thus eliminating divergences via G(k) = 0 and preventing negative energy states. In contrast, for proton or electron masses, R_m >> R_gp-GR (or R_gp), leading to E_T ~ Mc^2, rendering gravitational effects negligible and unsuitable for a cut-off. This affirms the Planck scale's unique role in quantum gravity.

If R_m > R_gp-GR, G(k) >0, yielding an attractive force. If R_m=R_gp-GR, G(k)=0, and the gravitational coupling vanishes. If R_m < R_gp-GR, G(k) < 0, yielding a repulsive force or antigravity. This repulsive force prevents singularity formation in black holes, thereby the singularity problem is also solved.

Edited May 4May 4 by icarus2

The Central Idea: Effective Mass and Running Gravitational Coupling G(k)

One of the key principles of General Relativity is that the energy-momentum tensor (T_μν) in Einstein's field equations already encompasses all forms of energy within a system, including rest mass, kinetic energy, and various binding energies. This implies that the mass serving as the source of gravity is inherently an 'effective mass' (M_eff), accounting for all such contributions, rather than a simple 'free state mass'. My paper starts from this very premise. By explicitly incorporating the negative contribution of gravitational self-energy into this M_eff, I derive a running gravitational coupling constant, G(k), that changes with the energy scale. This, in turn, provides a solution to long-standing problems in gravitational theory.

M_eff = M_fr − ∣U_gp∣/c^2

where M_fr is the free mass and U_gp is the gravitational self-energy (or binding energy).

From this concept of effective mass, I derive a running gravitational coupling constant, G(k). Instead of treating Newton's constant G_N as fundamental at all scales, my work shows that the strength of gravitational interaction effectively changes with the momentum scale k (or, equivalently, with the characteristic radius R_m of the mass/energy distribution). The derived expression, including general relativistic (GR) corrections for the self-energy, is:

1.Vanishing Gravitational Coupling and Resolution of Divergences

1)In Newtonian mechanics, the gravitational binding energy and the gravitational coupling constant G(k)

2)In the Relativistic approximation, the gravitational binding energy and the gravitational coupling constant G(k)

For R_m >>R_{gp-GR} ~ 0.58R_S, the gravitational self-energy term is negligible, and the running gravitational coupling G(k) returns to the gravitational coupling constant G_N.

As the radius approaches the critical value R_m = R_{gp-GR} ~ 0.58R_S, the coupling G(k) smoothly goes to zero, ensuring that gravitational self-energy does not diverge. Remarkably, this mechanism allows gravity to undergo self-renormalization, naturally circumventing the issue of infinite divergences without invoking quantum modifications.

For R_m < R_{gp-GR} ~ 0.58R_S, the gravitational coupling becomes negative (G(k)< 0), indicating a repulsive or antigravitational regime. This provides a natural mechanism preventing further gravitational collapse and singularity formation, consistent with the arguments in Section 2.

4.5. Solving the problem of gravitational divergence at high energy: Gravity's Self-Renormalization Mechanism

At low energy scales (E << M_Pc^2, Δt >>t_P), the divergence problem in gravity is addressed through effective field theory (EFT). However, at high energy scales (E ~ M_Pc^2, Δt~t_P), EFT breaks down due to non-renormalizable divergences, leaving the divergence problem unresolved.

Since the mass M is an equivalent mass including the binding energy, this study proposes the running coupling constant G(k) that reflects the gravitational binding energy.

At the Planck scale (R_m ~ R_{gp-GR} ~ 1.16(G_NM_fr/c^2) ~ l_P), G(k)=0 eliminates divergences, and on higher energy scales than Planck's (R_m < R_{gp-GR}), a repulsion occurs as G(k)<0, solving the divergence problem in the entire energy range. This implies that gravity achieves self-renormalization without the need for quantum corrections.

4.5.1. At Planck scale
If, M ~ M_P

R_{gp-GR} ~ 1.16(G_NM_P/c^2) = 1.16l_P

This means that R_{gp-GR}, where G(k)=0, i.e. gravity is zero, is the same size as the Planck scale.

4.5.2. At high energy scales larger than the Planck scale

In energy regimes beyond the Planck scale (R_m<R_{gp-GP}), where G(k) < 0, the gravitational coupling becomes negative, inducing a repulsive force or antigravity effect. This anti-gravitational effect prevents gravitational collapse and singularity formation while maintaining uniform density properties, thus mitigating UV divergences across the entire energy spectrum by ensuring that curvature terms remain finite.

4.5.3. Resolution of the two-loop divergence in perturbative quantum gravity via the effective mass framework

A crucial finding is that at a specific critical radius, R_{gp−GR}≈1.16(G_NM_fr/c^2) ≈ 0.58R_S (where R_S is the Schwarzschild radius based on M_fr), the negative gravitational self-energy precisely balances the positive free mass-energy. At this point, M_eff→0, and consequently, the effective gravitational coupling G(k)→0. This vanishing of the gravitational coupling has profound implications for quantum gravity. Perturbative quantum gravity calculations, which typically lead to non-renormalizable divergences (like the notorious 2-loop R^3 term identified by Goroff and Sagnotti), rely on the coupling constant κ=(32πG)^(1/2).
If G(k)→0 at high energies (Planck scale), then κ→0. As a result, all interaction terms involving κ diminish and ultimately vanish, naturally eliminating these divergences without requiring new quantum correction terms or exotic physics. Gravity, in this sense, undergoes a form of self-renormalization.

In perturbative quantum gravity, the Einstein-Hilbert action is expanded around flat spacetime using a small perturbation h_μν, with the gravitational field expressed as g_μν = η_μν+ κh_μν, where κ= \sqrt {32πG(k)} and G_N is Newton’s constant. Through this expansion, interaction terms such as L^(3), L^(4), etc., emerge, and Feynman diagrams with graviton loops can be computed accordingly.

At the 2-loop level, Goroff and Sagnotti (1986) demonstrated that the perturbative quantization of gravity leads to a divergence term of the form:

Γ_div^(2) ∝ (κ^4)(R^3)

This divergence is non-renormalizable, as it introduces terms not present in the original Einstein-Hilbert action, thus requiring an infinite number of counterterms and destroying the predictive power of the theory.

However, this divergence occurs by treating the mass M involved in gravitational interactions as a constant quantity. The concept of invariant mass pertains to the rest mass remaining unchanged under coordinate transformations; this does not imply that the rest mass of a system is intrinsically immutable. For instance, a hydrogen atom possesses different rest masses corresponding to the varying energy levels of its electrons. Both Newtonian gravity and general relativity dictate that the physically relevant source term is the equivalent mass, which includes not only rest mass energy but also binding energy, kinetic energy, and potential energy. When gravitational binding energy is included, the total energy of a system is reduced, yielding an effective mass:

M_eff = M_fr - M_binding

At this point R_m = R_{gp-GR} ~ 1.16(G_NM_fr/c^2), G(k) = 0, implying that the gravitational interaction vanishes.

As R_m --> R_{gp-GR}, κ= \sqrt {32πG(k)} -->0

Building upon the resolution of the 2-loop divergence identified by Goroff and Sagnotti (1986), our model extends to address divergences across all loop orders in perturbative gravity through the running gravitational coupling constant G(k). At the Planck scale (R_m=R_{gp-GR}), G(k)=0, nullifying the coupling parameter κ= \sqrt {32πG(k)} . If G(k) --> 0, κ --> 0.

As a result, all interaction terms involving κ, including the divergent 2-loop terms proportional to κ^{4} R^{3}, vanish at this scale. This naturally eliminates the divergence without requiring quantum corrections, rendering the theory effectively finite at high energies. This mechanism effectively removes divergences, such as the 2-loop R^3 term, as well as higher-order divergences (e.g., R^4, R^5, ...) at 3-loop and beyond, which are characteristic of gravity's non-renormalizability.

In addition, in the energy regime above the Planck scale (R_m<R_{gp-GR} ~ l_P), G(k)<0, and the corresponding energy distribution becomes a negative mass and negative energy state in the presence of an anti-gravitational effect. This anti-gravitational effect prevents gravitational collapse and singularity formation while maintaining uniform density properties, thus mitigating UV divergences across the entire energy spectrum by ensuring that curvature terms remain finite.

However, due to the repulsive gravitational effect between negative masses, the mass distribution expands over time, passing through the point where G(k)=0 due to the expansion speed, and reaching a state where G(k)>0. This occurs because the gravitational self-energy decreases as the radius R_m of the mass distribution increases, whereas the mass-energy remains constant at Mc^2. When G(k)>0, the state of attractive gravity acts, causing the mass distribution to contract again. As this process repeats, the mass and energy distributions eventually stabilize at G(k)=0, with no net force acting on them.

Unlike traditional renormalization approaches that attempt to absorb divergences via counterterms, this method circumvents the issue by nullifying the gravitational coupling at high energies, thus providing a resolution to the divergence problem across all energy scales. This effect arises because there exists a scale at which negative gravitational self-energy equals positive mass-energy.

~~~

2.Physical Origin of the Planck-Scale Cut-off

4.6. The physical origin of the cut-off energy at the Planck scale

In quantum field theory (QFT), the cut-off energy Λ or cut-off momentum is introduced to address the infinite divergence problem inherent in loop integrals, a cornerstone of the renormalization process. However, this cut-off has traditionally been viewed as a mathematical convenience, with its physical origin or justification remaining poorly understood.

This work proposes that Λ represents a physical boundary determined by the scale where the sum of positive mass-energy and negative gravitational self-energy equals zero, preventing negative energy states at the Planck scale. This mechanism, rooted in the negative gravitational self-energy of positive mass or energy, provides a physical explanation for the Planck-scale cut-off.

At R_m = R_{gp-GR} ~ 1.16(G_NM_fr/c^2), G(k)=0

For a mass M_fr ~ M_P, the characteristic radius is :

R_{gp-GR} ~ 1.16(G_NM_P/c^2) = 1.16l_P

At R_m=R_{gp-GR}, G(k)=0, marking the Planck scale where divergences vanish.

If R_m<R_{gp-GR}, then G(k)<0, which means that the system is in a negative mass state. Therefore, the Planck scale acts as a boundary energy where an object is converted to a negative energy state by the gravitational self-energy of the object. In a theoretical analysis, a negative mass state may be allowed, although the system can temporarily enter a negative mass state, the mass distribution expands again because there is a repulsive gravitational effect between the negative masses. Thus, the Planck scale (l_P) serves as a boundary preventing negative energy states driven by gravitational self-energy.

4.6.2. Uncertainty principle and total energy with gravitational self-energy

ΔEΔt≥hbar/2

ΔE≥hbar/2t_P=(1/2)M_Pc^2

During Planck time t_P, let's suppose that quantum fluctuations of (5/6)M_P mass have occurred.

Since all mass or energy is combinations of infinitesimal masses or energies, positive mass or positive energy has a negative gravitational self-energy. The total energy of the system, including the gravitational self-energy, is

E_T=Σm_ic^2 + Σ-Gm_im_j/r_ij = Mc^2 - (3/5)GM^2/R

Here, the factor 3/5 arises from the gravitational self-energy of a uniform mass distribution. Substituting (5/6)M_P and R=ct_P/2 (where cΔt represents the diameter of the energy distribution, constrained by the speed of light (or the speed of gravitational transfer). Thus, Δx = 2R= cΔt.

This demonstrates that at the Planck scale, the negative gravitational self-energy balances (or can be offset) the positive mass-energy, defining a cut-off energy Λ ~ M_Pc^2. For energies E>Λ, the system enters a negative energy state (E_T<0), which is generally prohibited due to the repulsive gravitational effects of negative mass states. Repulsive gravity prevents further collapse, dynamically enforcing the Planck scale as a minimal length.

This negative E_T indicates that R_m(= (1/2) l_P ) < R_{gp−GR}(= 1.16 l_P ), where R_{gp−GR} ∼ l_P is the critical radiusat which E_T = 0. Increasing ∆t ∼ t_p, R_m → R_{gp−GR}, and E_T → 0, suggesting that the Planck scale is wheregravitational self-energy can balance the mass-energy, supporting a physical cut-off at Λ ∼ M_Pc^2

The Planck scale exhibits a unique characteristic: only for M ~ M_P, t ~ t_P , and R ~ l_P does the gravitational self-energy (U_{gp-GR}) approach the mass-energy, enabling E_T ~ 0. This balance (or offset) suggests that the QFT cut-off Λ ~ M_Pc^2 acts as a physical boundary where quantum and gravitational effects converge. In contrast, for proton or electron masses, R_m >> R_{gp-GR}, rendering gravitational effects negligible and aligning with QED/QCD cut-offs (Λ ~ GeV).

3.Resolution of the Black Hole Singularity

For radii smaller than the critical radius, i.e., R_m<R_{gp−GR}, the expression for G(k) becomes negative (G(k)<0). This implies a repulsive gravitational force, or antigravity. Inside a black hole, as matter collapses, it would eventually reach a state where R_m<R_{gp−GR}. The ensuing repulsive gravity would counteract further collapse, preventing the formation of an infinitely dense singularity. Instead, a region of effective zero or even repulsive gravity would form near the center. This resolves the singularity problem purely within a gravitational framework, before quantum effects on spacetime structure might become dominant.

4.Cosmological Implications – A Potential Source for Dark Energy

The anti-gravitational regime (G(k)<0) predicted by my model has observable results on a cosmological scale. If the observable universe's average mass-energy distribution has an effective radius R_m that is less than its own critical radius R_{gp−GR}, then the universe itself would be in a state of repulsive gravitational self-interaction. This could provide a natural explanation for the observed accelerated expansion of the universe, attributing it to a fundamental property of gravity rather than an exotic dark energy component. My calculations suggest that for the estimated mass-energy of the observable universe, its current radius(R_m=46.5BLY) is indeed smaller than its R_{gp−GR}(275.7BLY), placing it in this repulsive regime.

The observable universe is in the R_m=46.5BLY < R_{gp-GR}=275.7BLY state, and therefore, an accelerated expansion exists.

Unifying Perspective

My research suggests that these seemingly disparate problems – gravitational divergences, the nature of the Planck scale cut-off, black hole singularities, and potentially even the mystery of dark energy – might share a common origin rooted in the proper accounting of gravitational self-energy. By incorporating this fundamental aspect of gravity, we can achieve a more consistent and predictive theory.

https://www.researchgate.net/publication/388995044_Solution_to_Gravity_Divergence_Gravity_Renormalization_and_Physical_Origin_of_Planck-Scale_Cut-off

Edited May 13May 13 by icarus2

On 5/14/2025 at 8:26 AM, icarus2 said:

2)In the Relativistic approximation, the gravitational binding energy and the gravitational coupling constant G(k)

For R_m >>R_{gp-GR} ~ 0.58R_S, the gravitational self-energy term is negligible, and the running gravitational coupling G(k) returns to the gravitational coupling constant G_N.

As the radius approaches the critical value R_m = R_{gp-GR} ~ 0.58R_S, the coupling G(k) smoothly goes to zero, ensuring that gravitational self-energy does not diverge. Remarkably, this mechanism allows gravity to undergo self-renormalization, naturally circumventing the issue of infinite divergences without invoking quantum modifications.

For R_m < R_{gp-GR} ~ 0.58R_S, the gravitational coupling becomes negative (G(k)< 0), indicating a repulsive or antigravitational regime. This provides a natural mechanism preventing further gravitational collapse and singularity formation, consistent with the arguments in Section 2.

2)In the Relativistic approximation, the gravitational binding energy and the gravitational coupling constant G(k)

For R_m >>R_{gp-GR} ~ 0.58R_S, the gravitational self-energy term is negligible, and the running gravitational coupling G(k) returns to the gravitational coupling constant G_N.

As the radius approaches the critical value R_m = R_{gp-GR} ~ 0.58R_S, the coupling G(k) smoothly goes to zero, ensuring that gravitational self-energy does not diverge. Remarkably, this mechanism allows gravity to undergo self-renormalization, naturally circumventing the issue of infinite divergences without invoking quantum modifications.

For R_m < R_{gp-GR} ~ 0.58R_S, the gravitational coupling becomes negative (G(k)< 0), indicating a repulsive or antigravitational regime. This provides a natural mechanism preventing further gravitational collapse and singularity formation, consistent with the arguments in Section 2.

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The concept of invariant mass pertains to the rest mass remaining unchanged under coordinate transformations; this does not imply that the rest mass of a system is intrinsically immutable. For instance, a hydrogen atom possesses different rest masses corresponding to the varying energy levels of its electrons. Both Newtonian gravity and general relativity dictate that the physically relevant source term is the equivalent mass, which includes not only rest mass energy but also binding energy, kinetic energy, and potential energy. When gravitational binding energy is included, the total energy of a system is reduced, yielding an effective mass:

M_eff = M_fr - M_binding

Any entity possessing spatial extent is an aggregation of infinitesimal elements. Since an entity with mass or energy is in a state of binding of infinitesimal elements, it already has gravitational binding energy or gravitational self-energy. And, this binding energy is reflected in the mass term to form the mass M_eff. It is presumed that the gravitational divergence problem and the non-renormalization problem occur because they do not consider the fact that M_eff changes as this binding energy or gravitational self-energy changes.

After writing the above explanation, additional applied research has been conducted.

In Chapter 4, it was shown that the divergence problem for two or more loops, as claimed by Goroff and Sagnotti, can be resolved by taking a κ=((32πG(k))^(1/2) → 0.

In Chapter 5, a solution to the divergence problem in the standard Effective Field Theory (EFT) proposed by John F. Donoghue and others is presented. In the conventional EFT model, although quantum correction terms exist,

calculations at or above the Planck scale reveal that these quantum correction terms are smaller than the General Relativity (GR) correction terms. This not only makes it difficult to verify quantum gravity effects but also leads to the breakdown of the EFT model near the Planck scale due to the divergence of GR correction terms.

In Chapter 5 of this paper, we not only address the divergence problem of GR correction terms near the Planck scale but also demonstrate that there exists a regime where GR correction terms are suppressed, becoming smaller than quantum gravity effects. In other words, a regime where quantum correction terms dominate exists, suggesting theoretical verifiability, even though technical verification is currently infeasible.

In Chapter 6, it is pointed out that the mainstream hypothesis—that the singularity problem inside black holes would be resolved by quantum mechanics—faces serious issues within the framework of standard EFT. Accordingly, this model should be more actively examined.

Chapter 5: Integration of Effective Field Theory (EFT) and the Running Coupling Constant G(k)

This chapter explains how the my model constructs a unified framework that complements and completes, rather than replaces, the standard tool of modern quantum gravity research: Effective Field Theory (EFT).

Existing EFT and its Limitations
Standard EFT treats general relativity as a valid quantum theory in the low-energy regime. The problem of high-energy divergences is handled by mathematically absorbing them into an infinite series of unknown coefficients, such as c_1R^2 and c_2R_μνR^μν, which parameterize our ignorance of high-energy physics. While this approach is successful for low-energy predictions, it is fundamentally limited by its inability to explain the high-energy phenomena themselves.

The Unified Model: Renormalization of the 'Gravitational Source'
I attempts to create a unified model by retaining the framework of EFT but redefining its most fundamental assumption: the source of gravity.

• Core Principle: The source of gravitational interaction is not the free-state mass (m_fr) but the effective mass (m_eff), which includes its own gravitational binding energy.

• Application Method: In the interaction potential formula derived from EFT, every mass term m is replaced with its effective counterpart m_eff.

• In this equation, m_eff is approximated as m_fr(1 - R_gs/R_m), which converges to zero as the object's radius R_m approaches the critical radius R_gs.

Result: The inversion is now far more pronounced. Quantum correction(~ 0.101) is over 10 times larger than the suppressed classical GR correction (~ 0.0096)

Result: The behavior diverges dramatically. The classical GR correction is not suppressed; instead, it grows to a value of ~ 0.971, indicating a breakdown of the perturbative expansion.

Physical Implications of the Unified Model
This simple substitution leads to dramatic differences in both low- and high-energy regimes.

• Consistency at Low Energies: For macroscopic objects (stars, planets, etc.), R_m>>R_gs, and therefore m_eff ~ m_fr, Consequently, the model's predictions perfectly align with those of standard EFT at low energies.

• The 'Master Switch' Effect at High Energies: As an object approaches its critical radius (R_m-->R_gs), its effective mass approaches zero (m_eff-->0). The global pre-factor m_1,eff * m_2,eff, which governs the entire potential, acts as a "master switch," simultaneously shutting down all interaction components: classical, relativistic, and even the quantum correction. This provides a fundamental resolution to the divergence problem.

At energies larger than the Planck scale, the standard EFT diverges.

Prediction of a 'Quantum-Dominant Regime': In standard EFT, the classical GR correction (proportional to mass) always dominates the quantum correction at high energies. In this model, however, the mass-dependent GR correction is suppressed by m_eff, while the relative importance of the quantum correction grows. This leads to the novel prediction of a "quantum-dominant regime" where quantum effects surpass classical effects just before gravity is turned off.

The existence of this regime is a direct consequence of treating the source mass as a dynamic entity that includes its own self-energy. It suggests that just before gravity 'turns itself off,' it passes through a phase where its quantum nature is maximally exposed. This provides, in principle, a unique experimental signature that could distinguish this self-renormalization model from standard EFT, should technology ever allow for probing physics at this scale.

Chapter 6: A New Paradigm for the Singularity Problem – A Gravitational Resolution, Not a Quantum One

This chapter argues why the mainstream hypothesis—that quantum mechanics resolves the black hole singularity—is difficult to sustain within the EFT framework, and proposes an alternative mechanism of "self-resolution by gravity."

Quantitative Rebuttal of the Mainstream Hypothesis

The mainstream view posits that at the Planck scale, unknown quantum gravity effects would generate a repulsive pressure to halt collapse. I tests this hypothesis quantitatively using the standard EFT framework.

Analysis of Correction Ratios: In standard EFT, the ratio between the classical GR correction (V_GR) and the quantum correction (V_Q) is derived as:

• V_GR/V_Q ≈ 4.66xy
  Here, x is the mass in units of Planck mass, and y is the distance in units of Planck length.

• The Case of a Stellar-Mass Black Hole: For the smallest stellar-mass black hole (3 solar masses), this ratio is calculated at the Planck length (y=1). The result is staggering:

  
  V_GR/V_Q ≈ 4.66×(2.74×10^38)×1 ≈ 1.28×10^39

  
  This demonstrates that under the very conditions where quantum effects are supposed to become dominant, the classical GR effect overwhelms the quantum effect by a factor of ~10³⁹. As a result of analysis by the standard EFT model, it is therefore likely that there is a problem with the mainstream speculation that quantum effects will provide the repulsive force needed to solve the singularity.

The Gravitational Solution: A Paradigm Shift
I claims the solution to the singularity problem is not quantum mechanical, but is already embedded within general relativity itself.

• The Agent of Resolution: The force that halts collapse is not quantum pressure but a gravitational repulsive force that arises when m_eff becomes negative in the region R_m < R_gs.

• The Scale of Resolution: This phenomenon occurs not at the microscopic Planck length (~10⁻³⁵ m) but at the macroscopic critical radius R_gs, which is proportional to the black hole's Schwarzschild radius(R_S), specifically R_gs ~ G_NM_fr/c^2 ~ 0.5R_S. This means that even for the smallest stellar-mass black holes, collapse is halted at a scale of several kilometers.

In conclusion, the paper proposes a paradigm shift: the singularity is not resolved by quantum mechanics "rescuing" general relativity, but rather by gravity resolving its own issue. The mechanism is purely gravitational and operates on a macroscopic scale, well before quantum effects could ever become relevant.

#Paper

Solution to Gravity Divergence, Gravity Renormalization, and Physical Origin of Planck-Scale Cut-off

Edited June 15Jun 15 by icarus2